Parabolic Arcs Research Articles

PYTHAGOREAN PARABOLOGRAPH FOR REGIONS’ QUADRATURE

<p>In this paper an attempt is made to design a mathematical machine based on the Pythagorean theory of fitting an equivalent parallelogram (given angle and side) with another parallelogram. An equivalent square or rhombus is constructed given a rectangle or a parallelogram respectively. It is then proved that this mechanism can simultaneously draw a parabolic arc. The parameters of this mathematical machine are checked. In each case, an attempt is made to find the focus and the directrix of the corresponding parabola. The important help of the GeoGebra software for the initial mechanical design of the machine is highlighted, but also the important differences of this simulation from the physical construction.</p><p> </p><p><strong> Article visualizations:</strong></p><p><img src="/-counters-/soc/0713/a.php" alt="Hit counter" /></p>

An efficient finite element computation using subparametric transformation up to cubic-order for curved triangular elements

PurposeA finite element computational methodology on a curved boundary using an efficient subparametric point transformation is presented. The proposed collocation method uses one-side curved and two-side straight triangular elements to derive exact subparametric shape functions.Design/methodology/approachOur proposed method builds upon the domain discretization into linear, quadratic and cubic-order elements using subparametric spaces and such a discretization greatly reduces the computational complexity. A unique subparametric transformation for each triangle is derived from the unique parabolic arcs via a one-of-a-kind relationship between the nodal points.FindingsThe novel transformation derived in this paper is shown to increase the accuracy of the finite element approximation of the boundary value problem (BVP). Our overall strategy is shown to perform well for the BVP considered in this work. The accuracy of the finite element approximate solution increases with higher-order parabolic arcs.Originality/valueThe proposed collocation method uses one-side curved and two-side straight triangular elements to derive exact subparametric shape functions.

Fast permeability measurement for tight reservoir cores using only initial data of the one chamber pressure pulse decay test

In this study, a mathematical model for fast determination of the permeabilities of tight rocks using measurements taken from the initial period of the One Chamber Pressure Pulse Decay (OC-PPD) test is presented. The model applies to measurements taken both before and after the pressure pulse front has reached the downstream end of the specimen. The analytical solutions for the pressure decay in the upstream chamber are derived based on a parabolic arc approximation of pore pressure distribution along the test specimen. This approximation allows converting the initial–boundary value problem of fluid diffusion in the specimen, governed by partial differential equations, to a system of ordinary differential equations that can be easily solved by explicit formulae. Thus, an explicit formula for the pressure decay rate is obtained, which enables inverse analysis of the initial experimental data to estimate the rock permeability. The proposed method expedites the pulse decay test as it does not require the system to reach equilibrium. The method is validated with three sets of experimental data of the OC-PPD test using helium as the diffusing fluid, for which the relative error of the permeability is found to be less than 6%. This method is particularly useful if the equilibrium time of the pulse decay test for rock specimens with permeabilities in the range of nano-Darcy takes hours or days.

Probing the Interstellar Medium from Scintillation of the Swooshing Pulsar B0919+06

Radio observations of pulsars offer a potential method to probe the intricate microstructure in the turbulent interstellar medium. Here we report on a high-resolution dynamic spectral analysis of the “swooshing pulsar” B0919+06 observed with the Five-hundred-meter Aperture Spherical radio Telescope over multiple epochs and with the ultrawideband receiver on the Parkes radio telescope. For all observations, the dynamic scintillation spectra, two-dimensional autocovariance functions, and secondary spectra are presented. At 1250 MHz, the decorrelation bandwidth, diffraction timescale, and the drift rate are determined to be Δν d = 25.89 ± 7.55 MHz, Δτd=14.42±3.98 minutes, and dt/dν=0.07±0.14minutes MHz−1, respectively. The frequency dependencies of the scintillation parameters exhibit single power-law spectral behaviors, indicating that the electron density fluctuations in the interstellar medium approximately follow the Kolmogorov spectrum. The secondary spectra exhibit two distinct parabolic arcs with well-determined curvatures of 0.002 and 0.02 s3 for the outer and inner arcs, respectively. The locations of the scattering screens are approximately determined to be 157.3 and 726.0 pc, respectively, from the pulsar for isotropic scattering. The inner scintillation arc is present contemporaneously over a wide frequency range, indicating that the scintillation arc is a broadband phenomenon. The arc curvature scales with observing frequency as a power law with an index of −2.05 ± 0.05, which implies that the scattering screen spans a physical distance from 689.7 to 883.3 pc from the pulsar.

Scattering model of scintillation arcs in pulsar secondary spectra

ABSTRACT The dynamic spectra of pulsars frequently exhibit diverse interference patterns, often associated with parabolic arcs in the Fourier-transformed (secondary) spectra. Our approach differs from previous ones in two ways: first, we extend beyond the traditional Fresnel–Kirchhoff method by using the Green’s function of the Helmholtz equation, i.e. we consider spherical waves originating from three-dimensional space, not from a two-dimensional screen. Secondly, the discrete structures observed in the secondary spectrum result from discrete scatterer configurations, namely plasma concentrations in the interstellar medium, and not from the selection of points by the stationary phase approximation. Through advanced numerical techniques, we model both the dynamic and secondary spectra, providing a comprehensive framework that describes all components of the latter spectra in terms of physical quantities. Additionally, we provide a thorough analytical explanation of the secondary spectrum.

Numerical Solution of Natural Frequencies and Mode Shapes

Every construction has a set of their natural frequencies (resonant frequencies) and mode shapes that depend on its material, structure, and boundary conditions. A mode shape is a deflection pattern. It is related to a particular natural frequency and represents the relative displacement of all parts of a construction for that particular mode. This paper deals with numerical solutions of natural frequencies and mode shapes. The method of stiffness constants and bisection method is used. The procedure can be used for any planar bar construction. First, the mode shapes of the simple beam are solved, where the solution is compared with the results obtained using known relationships. The article further describes the solution of the parabolic arc.

ON THE MAXIMUM CIRCULAR INVERSES OF MAXIMUM CIRCLES

In this study, the inverses of maximum circles under the maximum circular inversion are examined. The maximum circular inversion is observed to transform maximum circles to curves consisting of parabolic arcs, line segments, and sometimes rays. It is seen that the image curve occurs in different shapes depending on the relative positions of the inversion circle and the maximum circle. While the images of maximum circles not passing through the inversion center are closed curves, the images of those passing through the inversion center are not closed. Inverse curves have been studied by considering the radii of the maximum circles and the positions of the lines joining their centers and the center of maximum inversion. Classifications of the inverses of maximum circles are presented.

Finding the Weight Difference of a Rectangular Structure with a Parabolic Arc for Mathematical Models Made of Steel and other Composite Materials

The most popular materials for lightweight constructions, including building and aircraft structures, industrial, military, and aerospace technology, are armored composites. Composites made of carbon fiber are typically employed in lightweight applications. The ANSYS program was used to produce four mathematical models. Steel is used in the construction of the first and second versions whereas composite materials are used in the third and fourth variants. To find all the deformations, stresses, and strains that appear on the four models, as well as to calculate the weights of those four structures and compare them, these four models were tested with the ANSYS 15.0 program to obtain equal deformation resistance for all models under the influence of different loads. The results show that the composite models had lower strains, stresses, and deformations than the steel models. Among other results, it was discovered that the weight of the third model made of composite materials decreased by (32.72%) compared to the steel-based first model, and after doing the necessary calculations and assessing the results, the fourth model made of composite materials' weight was reduced by (19.21%) when compared to the second model made of steel.

Interstellar scintillation observations for PSR J0835−4510 at 6656 MHz

ABSTRACT Scintillation phenomena of pulsars provide an important tool for studying interstellar scattering screens and their properties. In this paper, we report pulsar scintillation phenomena of the PSR J0835−4510 at multiple epochs between MJD 58592 and MJD 58843 using the Kunming 40 m radio telescope at a centre frequency of 6656 MHz. We have measured the dynamic spectrum, and derived the scintillation time, bandwidth, and secondary spectrum at each observation epoch. For the first time, we report the detection of parabolic arc structure in the secondary spectra of PSR J0835−4510. Parabolic arcs with varying curvatures appear at all our observation epochs. We used both anisotropic and isotropic scattering screen models to fit the annual variations of arc curvature. While both models can fit the arc curvature variations equally well, the anisotropic model also explains the parabolic arc morphology and its time variation. In this model, the scattering is located at a fractional distance of s = 0.35 ± 0.10, or a distance Ds = 0.19 ± 0.03 kpc from the Earth, likely corresponding to the boundary of the Local Bubble.

Birational Quadratic Planar Maps with Generalized Complex Rational Representations

Complex rational maps have been used to construct birational quadratic maps based on two special syzygies of degree one. Similar to complex rational curves, rational curves over generalized complex numbers have also been constructed by substituting the imaginary unit with a new independent quantity. We first establish the relationship between degree one, generalized, complex rational Bézier curves and quadratic rational Bézier curves. Then we provide conditions to determine when a quadratic rational planar map has a generalized complex rational representation. Thus, a rational quadratic planar map can be made birational by suitably choosing the middle Bézier control points and their corresponding weights. In contrast to the edges of complex rational maps of degree one, which are circular arcs, the edges of the planar maps can be generalized to hyperbolic and parabolic arcs by invoking the hyperbolic and parabolic numbers.

Pressure and flow transition over NACA airfoil with thrust optimized parabolic arcs

This paper presents an accurate prediction of surface drag, pressure coefficient and flow transition around a symmetric airfoil design through the implementation of finite element discretization. Skin friction coefficient has been analyzed at different Reynolds number and at various angles of attack for NACA0012 airfoil design. The transition from laminar to turbulent flow significantly impacts the separation of the boundary layer and skin friction of the airfoil, ultimately affecting its aerodynamic characteristics. The pressure distribution around the surface of NACA0012 has also been computed at several Reynolds number and different attack angles using higher order triangular meshes. Numerical evaluation of these results when compared with the experimental results demonstrates superior accuracy with higher-order finite element mesh. The investigation carried out by higher-order element meshing approach is based on the subparametric transformation of parabolic arcs of triangular element curved at one side. Moreover, the high aspect ratio meshes obtained during discretization of airfoil design have been considered for the analysis. The investigation has yielded results that closely align with experimental data, demonstrating excellent performance. The current efficient analysis of pressure and flow transition is beneficial in interpreting aerodynamic performance and flow simulations for aerospace and computational fluid dynamics applications.

Influence of process parameters on single weld seam geometry and process stability in Laser Hot-Wire Cladding of AISI 52100

Steels with high carbon content can hardly or not at all be welded, but are of great interest for cladding applications due to their high hardness. In this study, the influence of process parameters on weld seam geometry and process stability is investigated when welding AISI 52100 bearing steel using the laser hot-wire cladding process. Process stability is evaluated using actual and set values for the wire feed rate and current parameters to determine a process window for a stable welding process. Weld seams are measured and analyzed in terms of width, height, contact angle, and shape. The effect of the process parameters on the weld seam geometry is investigated and appropriate mathematical functions to describe the geometry are determined. Process parameter sets in the range of 1-2 m/min wire feed rate and 45-75 A hot wire current were investigated. Unstable parameter sets occur clustered at high wire feed rate of 2 m/min for all hot wire currents. In addition, the process is unstable at high hot wire current of 75 A and low wire feed speed of 1 m/min. The remaining parameter sets resulted in a stable process. The investigated functions parabolic, cosinusoidal and circular arc for the mathematical description of the weld seam geometry, no clearly significant result could be determined. Only a trend towards the circular arc function and the parabolic function is apparent.

Cubic Trigonometric Hermite Interpolation Curve: Construction, Properties, and Shape Optimization

Cubic Hermite interpolation curve plays a very important role in interpolation curves modeling, but it has three shortcomings including low continuity, difficult shape adjustment, and the inability to accurately represent some common engineering curves. We construct a cubic trigonometric Hermite interpolation curve to make up the three shortcomings of cubic Hermite interpolation curve once and for all. The cubic trigonometric Hermite interpolation curve not only inherits the features of cubic Hermite interpolation curve but also achieves C2 continuity, has local and global adjustability, and can accurately represent elliptical arc, circular arc, quadratic parabolic arc, cubic parabolic arc, and astroid arc that often appear in engineering. In addition, we give the schemes for optimizing the shape of the cubic trigonometric Hermite interpolation curve based on internal energy minimization. The schemes include optimizing the shape of planar curve and spatial curve. Some modeling examples show that the proposed schemes are effective and the cubic trigonometric Hermite interpolation curve is more practical than cubic Hermite interpolation curve.

The Thousand Pulsar Array programme on MeerKAT – X. Scintillation arcs of 107 pulsars

ABSTRACT We present the detection of 107 pulsars with interstellar scintillation arcs at 856–1712 MHz, observed with the MeerKAT Thousand Pulsar Array Programme. Scintillation arcs appear to be ubiquitous in clean, high S/N observations, their detection mainly limited by short observing durations and coarse frequency channel resolution. This led the survey to be sensitive to nearby, lightly scattered pulsars with high effective velocity – from a large proper motion, a screen nearby the pulsar, or a screen near the Earth. We measure the arc curvatures in all of our sources, which can be used to give an estimate of screen distances in pulsars with known proper motion, or an estimate of the proper motion. The short scintillation time-scale in J1731−4744 implies a scattering screen within 12 pc of the source, strongly suggesting the association between this pulsar and the supernova remnant RCW 114. We measure multiple parabolic arcs of five pulsars, all of which are weakly scintillating with high proper motion. Additionally, several sources show hints of inverted arclets suggesting scattering from anisotropic screens. Building on this work, further targeted MeerKAT observations of many of these pulsars will improve understanding of our local scattering environment and the origins of scintillation; annual scintillation curves would lead to robust screen distance measurements, and the evolution of arclets in time and frequency can constrain models of scintillation.

Interstellar Scintillation and Polarization of PSR B0656+14 in the Monogem Ring

High-sensitivity interstellar scintillation and polarization observations of PSR B0656+14 made at three epochs over a year using the Five-hundred-meter Aperture Spherical radio Telescope (FAST) show that the scattering is dominated by two different compact regions. We identify the one nearer to the pulsar with the shell of the Monogem Ring, thereby confirming the association. The other is probably associated with the Local Bubble. We find that the observed position angles of the pulsar spin axis and the spatial velocity are significantly different, with a separation of 19.°3 ± 0.°8, inconsistent with a previously published near-perfect alignment of 1° ± 2°. The two independent scattering regions are clearly defined in the secondary spectra, which show two strong forward parabolic arcs. The arc curvatures imply that the scattering screens corresponding to the outer and inner arcs are located approximately 28 pc from PSR B0656+14 and 185 pc from the Earth, respectively. Comparison of the observed Doppler profiles with electromagnetic simulations shows that both scattering regions are mildly anisotropic. For the outer arc, we estimate the anisotropy A R to be approximately 1.3, with the scattering irregularities aligned parallel to the pulsar velocity. For the outer arc, we compare the observed delay profiles with delay profiles computed from a theoretical strong-scattering model. Our results suggest that the spatial spectrum of the scattering irregularities in the Monogem Ring is flatter than Kolmogorov, but further observations are required to confirm this.

Wave attenuation and focusing performance of parallel twin parabolic arc floating breakwaters

The hybrid system consisting of floating breakwater and point absorber wave energy converters provides a promising solution for shoreline protection and wave power generation. In the hybrid system, the breakwater plays an important role in protecting the sheltered area on the lee side and focusing high waves for better energy harvesting on the weather side. To improve the wave attenuation and focusing performance, a twin-breakwater consisting of a pair of parallel parabolic pontoons is proposed. Based on the potential flow theory of linear waves, the influences of gap width and connection method applied between the two pontoons are studied in the frequency domain, with an emphasis on the so-called critical mode around which both wave attenuation and focusing could be improved. Results show that the rigidly connected twin-breakwater is superior to the unconnected twin-breakwater with the same configuration in both wave attenuation and focusing. A second critical mode with lower frequency is also found under particular gap width, providing a potential for the defense of long waves. An optimal attenuation could be obtained by applying a proper gap width.

Dividing the Perimeter of a Triangle into Unequal Proportions

We fully describe the envelope of all line segments that divide the perimeter of a triangle into the ratio α : 1 − α as α varies from 0 to 1 / 2 . If α is larger than the ratio of the longest side length to the perimeter, then the envelope is a 12-sided closed curve consisting of six line segments and six parabolic arcs. For other values of α , the envelope is the union of one to three parabolic arcs and possibly a 5- or 9-sided nonclosed curve consisting of line segments and parabolic arcs.

Geometric approximation scheme for parabolic arcs

A cubic H-Bézier geometric approximation scheme with one free parameter τ is introduced for parabolic arcs. It has four control points. The control points are computed by matching the end points and end unit tangents of the two curves and provide two more free parameters. These free parameters are computed by curvature continuity constraints. Optimal value of τ is obtained by minimizing the maximum value of the distance between the cubic H-Bézier curve and the concerned parabolic arc. The method is illustrated using different numerical examples which show that the proposed method is economical, reliable and efficient.

Empirical Model for the Description of Weld Seam Geometry in Coaxial Laser Hot-Wire Deposition Welding Processes with Different Steel Wires

Claddings are used to protect areas of components that are exposed to particular chemical, physical or tribological stresses. The aim when developing a cladding process is to achieve a cladding with low waviness in order to reduce the amount of machining required. Computational models and FEA simulations can be used to determine process parameters for claddings with low rework including a prediction of the height and width of a single weld seam aswell as the development of welding strategies. In this paper empirical models describing the geometry of single weld seams on a substrate manufactured with a coaxial laser hot-wire cladding process are investigated for three steel wire materials and different welding parameters. The coordinates of surface points of the weld seams were detected using a laser scanning microscope and post-processed by a self-created script. In order to describe the cross sectional shape of the weld seams, the parameters of parabolic, cosinusiodal or circular arc model functions are derived from the surface data using a fitting algorithm. For the tested wire materials, an effect of the wire material on the shape of the weld seam was not observed. The investigations also show that regardless of the varied welding parameter set or wire material, a circular model function appears to be the most suitable model shape for describing the cross sectional weld seam geometry in coaxial laser metal deposition with hot-wire. The regression residua using a circular arc model function ranged from 18.9 upmum to 34.6 upmum, which indicates a good approximation.

On big pieces approximations of parabolic hypersurfaces

Let \(\Sigma\) be a closed subset of \(\mathbb{R}^{n+1}\) which is parabolic Ahlfors-David regular and assume that \(\Sigma\) satisfies a 2-sided corkscrew condition. Assume, in addition, that \(\Sigma\) is either time-forwards Ahlfors-David regular, time-backwards Ahlfors-David regular, or parabolic uniform rectifiable. We then first prove that \(\Sigma\) satisfies a weak synchronized two cube condition. Based on this we are able to revisit the argument of Nyström and Strömqvist (2009) and prove that \(\Sigma\) contain suniform big pieces of Lip(1,1/2) graphs. When \(\Sigma\) is parabolic uniformly rectifiable the construction can be refined and in this case we prove that \(\Sigma\) contains uniform big pieces of regular parabolic Lip(1,1/2) graphs. Similar results hold if \(\Omega\subset\mathbb{R}^{n+1}\) is a connected component of \(\mathbb{R}^{n+1}\setminus\Sigma\) and in this context we also give a parabolic counterpart of the main result of Azzam et al. (2017) by proving that if \(\Omega\) is a one-sided parabolic chord arc domain, and if \(\Sigma\) is parabolic uniformly rectifiable, then \(\Omega\) is in fact a parabolic chord arc domain. Our results give a flexible parabolic version of the classical (elliptic) result of David and Jerison (1990) concerning the existence of uniform big pieces of Lipschitz graphs for sets satisfying a two disc condition.